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Properties

Label 16.0.33182682172...4413.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{10}\cdot 53^{7}$
Root discriminant $19.17$
Ramified primes $7, 53$
Class number $1$
Class group Trivial
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 0, 18, 42, -27, 16, -49, -5, 49, 16, 27, 42, -18, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 18*x^13 + 42*x^12 + 27*x^11 + 16*x^10 + 49*x^9 - 5*x^8 - 49*x^7 + 16*x^6 - 27*x^5 + 42*x^4 + 18*x^3 + 3*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 - 18*x^13 + 42*x^12 + 27*x^11 + 16*x^10 + 49*x^9 - 5*x^8 - 49*x^7 + 16*x^6 - 27*x^5 + 42*x^4 + 18*x^3 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 18 x^{13} + 42 x^{12} + 27 x^{11} + 16 x^{10} + 49 x^{9} - 5 x^{8} - 49 x^{7} + 16 x^{6} - 27 x^{5} + 42 x^{4} + 18 x^{3} + 3 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(331826821728530394413=7^{10}\cdot 53^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.17$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{10} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{3}{32} a^{9} - \frac{1}{16} a^{8} - \frac{3}{32} a^{7} + \frac{3}{32} a^{6} - \frac{13}{32} a^{5} - \frac{3}{16} a^{4} + \frac{7}{32} a^{3} + \frac{11}{32} a^{2} + \frac{7}{32} a - \frac{11}{32}$, $\frac{1}{32} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{3}{16} a^{7} + \frac{7}{32} a^{5} - \frac{3}{32} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{7}{16} a - \frac{5}{32}$, $\frac{1}{8384} a^{14} - \frac{7}{4192} a^{13} - \frac{107}{8384} a^{12} + \frac{97}{8384} a^{11} - \frac{173}{4192} a^{10} + \frac{1}{16} a^{9} - \frac{165}{4192} a^{8} - \frac{1561}{8384} a^{7} + \frac{279}{2096} a^{6} + \frac{1}{4} a^{5} + \frac{21}{2096} a^{4} + \frac{1669}{8384} a^{3} + \frac{1155}{8384} a^{2} + \frac{31}{1048} a - \frac{2359}{8384}$, $\frac{1}{8384} a^{15} - \frac{41}{8384} a^{13} - \frac{91}{8384} a^{12} - \frac{149}{4192} a^{11} + \frac{67}{4192} a^{10} - \frac{427}{4192} a^{9} - \frac{679}{8384} a^{8} - \frac{5}{1048} a^{7} + \frac{345}{4192} a^{6} - \frac{317}{1048} a^{5} - \frac{3705}{8384} a^{4} + \frac{3299}{8384} a^{3} - \frac{939}{2096} a^{2} + \frac{1375}{8384} a - \frac{1317}{4192}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13226.1412134 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.2597.2, 8.0.357453677.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{6}$ $16$ $16$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$53$53.4.3.4$x^{4} + 424$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$